R Exercise 4. Clustering and classification
I have loaded the Boston dataset that contains information on housing values in suburbs of Boston. Data has 5-6 observation and 14 differeent variables like crime rate of the twon, number of rooms in dwelling,pupil-teacher ration, proportion of lower status population, etc. All these variables are the key that help to eveluate the value of the houses in this area.
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...## [1] 506 14The graphical repreentation of the variables show that in some cases there exist a strong correltion between varibales; also the accumaulation tends to be close to the edge.
## function (x, ...)
## UseMethod("pairs")
## <bytecode: 0x7f81f32bbd18>
## <environment: namespace:graphics>## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00However, I will also plot the correlation matrix in order to explore the data in more details.
## Warning: package 'tidyverse' was built under R version 3.3.2## Loading tidyverse: ggplot2
## Loading tidyverse: tibble
## Loading tidyverse: tidyr
## Loading tidyverse: readr
## Loading tidyverse: purrr
## Loading tidyverse: dplyr## Warning: package 'ggplot2' was built under R version 3.3.2## Warning: package 'tidyr' was built under R version 3.3.2## Conflicts with tidy packages ----------------------------------------------## filter(): dplyr, stats
## lag(): dplyr, stats
## select(): dplyr, MASS## crim zn indus chas nox rm age dis rad tax
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47
## ptratio black lstat medv
## crim 0.29 -0.39 0.46 -0.39
## zn -0.39 0.18 -0.41 0.36
## indus 0.38 -0.36 0.60 -0.48
## chas -0.12 0.05 -0.05 0.18
## nox 0.19 -0.38 0.59 -0.43
## rm -0.36 0.13 -0.61 0.70
## age 0.26 -0.27 0.60 -0.38
## dis -0.23 0.29 -0.50 0.25
## rad 0.46 -0.44 0.49 -0.38
## tax 0.46 -0.44 0.54 -0.47
## ptratio 1.00 -0.18 0.37 -0.51
## black -0.18 1.00 -0.37 0.33
## lstat 0.37 -0.37 1.00 -0.74
## medv -0.51 0.33 -0.74 1.00
The correlogram is giving a more comprehensive picture of the coreelations between the variables. Therefore, one can clearly observe a negative correlation between indus/dis (proportion of non-retail business acres per town to weighted mean of distances to five Boston employment centres), nox/dis (nitrogen oxides concentration to weighted mean of distances to five Boston employment centres), age/dis (proportion of owner-occupied units built prior to 1940 to weighted mean of distances to five Boston employment centres) and lstat/medv (lower status of the population to median value of owner-occupied homes). Positive correlation is observed in indus/nox (proportion of non-retail business acres per town to nitrogen oxides concentration), rad/tax (index of accessibility to radial highways to full-value property-tax rate per $10,000), nox/age. Having a closer look at the correlations, these seem to be logical.
However, in order to be able to classify the variable, it has to be standirdized so that it is comparable.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865## [1] "matrix"Standardization of the variable have led to the fact that the range of the variable have decrease. Therefore, these standardized variables will be used in the further analysis
Also, I will create yet another categoriacl variable crime that will be created from the continuous one crim. I will remove crim variable from the dataset so that it does not affect the further analysis
## crime
## low med_low med_high high
## 1784 1854 1675 1771## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discardedAfter the necessary transformation, I will divide the data by the train (contain 80% of the data) and the test one (20% of the data) in order to proceed with the Linear Discrimination Analysis.
## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discardedNow, I will fit the linear discriminant analysis on the train set.I will use the categorical crime rate as the target variable and all the other variables in the dataset as predictor variables. LDA will fid a combination of the explanatory variable in such way so that it can separate the classes of the crime variable the best
## [1] "matrix"## crime
## low med_low med_high high
## 1784 1854 1675 1771## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discarded## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2541027 0.2627493 0.2350450 0.2481031
##
## Group means:
## zn indus chas nox rm
## low 0.24825158 -0.24340253 -0.01532391 -0.2261148 0.10701407
## med_low -0.09325430 -0.15077666 -0.07401979 -0.1210213 -0.01790638
## med_high -0.11209743 -0.02969243 0.03507215 -0.0957422 -0.02303700
## high -0.08486296 0.44415425 0.06929698 0.4687796 -0.06158741
## age dis rad tax ptratio
## low -0.25651713 0.22937984 -0.2281897 -0.2499182 -0.225008350
## med_low -0.03568832 0.05231709 -0.1906708 -0.1855142 0.002298817
## med_high -0.06551581 0.03529415 -0.1828589 -0.1556714 -0.031407746
## high 0.38408606 -0.33543923 0.6087474 0.6006695 0.238350986
## black lstat medv
## low 0.02052373 -0.15301093 0.15569204
## med_low 0.09373194 -0.08531061 0.01445482
## med_high 0.16548201 -0.13614641 0.02514190
## high -0.33290858 0.37593407 -0.20536690
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.42213641 0.73413008 -0.08391949
## indus 0.15436624 0.02744157 -1.42943840
## chas 0.07603136 -0.01181330 -0.35080816
## nox 0.24203818 0.07430928 0.28917581
## rm 0.17806697 0.09769646 -0.06397810
## age 0.17154013 -0.42563523 0.48406239
## dis 0.12993228 -0.06623937 -0.10790381
## rad 0.54586604 0.33856140 -0.36477229
## tax 0.18870520 -0.52247390 0.34604985
## ptratio 0.13795189 -0.19436834 0.44188529
## black -0.04277270 -0.36795929 -0.30821605
## lstat 0.36255746 0.62336005 0.46237427
## medv 0.26216018 0.30515033 0.35204524
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.8046 0.1566 0.0388
In order to see the full picture of the obtained results, I have plotted a graph where different colours are resposible for different clsses of the variables. The arrow indicates the impact of each of the predictor variable in the model.
Now, I will remove the crime from the data and will make a prediction for the new dataset.
## crime
## low med_low med_high high
## 1784 1854 1675 1771## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discarded## predicted
## correct low med_low med_high high
## low 109 121 48 63
## med_low 83 169 66 62
## med_high 75 129 54 59
## high 76 57 42 204The data show that prediction for the high crime rates are correct ones. However, the prediction for the low and medium crime rates are not always correct
Analysing the data from another angel, I will cluster observation and perform the k-means model that will asign cluster based on the distance between variables. Distance between the variablesis a measure of its similarity.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.119 85.620 170.500 226.300 371.900 626.000
- When performing k-means model, I have set the number of the clusters equal to 10. Howevr, from the plot (Within Cluster Sum of Squares), it obvious that 2 cluster will enough due to the fact that after two clusters, the value drops significantly. So, I will adjust the model.*
The plot is showing the scaled pairs that are lotted against each other
- Also, I have decided to make a 3D plot I have fit the scaled train data that you used to fit the LDA.*
## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discarded## predicted
## correct low med_low med_high high
## low 134 111 41 73
## med_low 64 189 45 74
## med_high 61 120 58 80
## high 72 53 41 201## [1] 5667 13## [1] 13 3## Warning: package 'plotly' was built under R version 3.3.2##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutAlso, here is another 3D plot where the color is defined by the clusters of the k-mean. It shows the same classes as in LDA model, howver, eithout classification by the crime rate.
## Warning in data.frame(boston_scaled, crime): row names were found from a
## short variable and have been discarded## predicted
## correct low med_low med_high high
## low 118 107 54 52
## med_low 73 182 58 63
## med_high 78 131 56 57
## high 72 60 31 225